A review of: “FUZZY SETS,UNCERTAINTY, AND INFORMATION,” by George J. Klir and Tina A. Folger. Prentice Hall,Englewood Cliffs,N.J., 1988, XI+ 355 pp.

Possibility distribution introduced by Zadeh "Fuzzy sets as a basis for a theory of possibility theory", Fuzzy Sets Syst. 1 (1978) 3-28] in his introductory paper of possibility theory assumes a normal distribution, in the sense that it supposes the existence of at least one element s 0 of the universe of discourse U , for which the distribution ~ is fully compatible with the context of interest: ~ (s 0 )=1. However, when such element does no longer exist, it leads to a subnormal possibility distribution. This situation may arise from incomplete data, inconsistent statements, or contradictory beliefs. To deal with such case, many authors like Yager "On the relationships of methods of aggregation evidence in expert systems", Cybern. Syst. , 16 (1985) 1-21; "A modification of the certainty measure to handle subnormal distributions", Fuzzy Sets Syst. , 20 (1986) 317-324], Dubois and Prade "An alternative approach to the handling of subnormal possibility distributions--A critical comment on a proposal of Yager", Fuzzy Sets Syst. , 24 (1987) 123-126] have put forward some proposals in order to keep track of the consistency of the basic axioms attached to possibility and necessity measures. In this paper, the proposals are reviewed in the light of new results regarding some appealing criteria. Particularly, when subnormal distribution and normal distribution are encountered in the same level, intuitively, two approaches are possible: Either the subnormal distributions are risen up to a normal distribution level, or the normal ones are flatted down to agree with the normal ones. In both cases there is a sort of gaining or losing information. We review some of the proposal solutions. The flatting approach is mainly related to fuzzy arithmetic calculus while the rising effect is motivated by Dempster-Shafer theory of evidence and its normalization paradigm. The two approaches will also be investigated with respect to some appealing criteria like preference preservation, distance minimization, entropy, minimum/maximum specificity, and, further, particular interest is focused on information based uncertainty preservation. Later on, the proposals are discussed according to the f -certainty qualification where the greatest value h of the subnormal distribution is understood as a degree of certainty that must be attached to the resulting normal distribution.